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Gait and Posture 15 (2002) 220– 229
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A model for lever-arm length calculation of the flexor and
extensor muscles at the ankle
Alberto Leardini a,*, John J. O’Connor b
a
Mo6ement Analysis Laboratory, Istituti Ortopedici Rizzoli, Via di Barbiano 1 /10, 40136 Bologna, Italy
b
Oxford Orthopaedic Engineering Centre, Uni6ersity of Oxford, Oxford, UK
Received 7 November 2000; received in revised form 4 June 2001; accepted 14 June 2001
Abstract
A sagittal-plane mathematical model of joint mobility, including the mechanical effect of the extensor retinacula, was used to
predict the lever arm lengths of the main flexor and extensor muscles of the human ankle over the range of movement. In
plantarflexion, the centre of rotation lies posteriorly and distally, maximising the lever arm of the tibialis anterior. The action of
the gastrocnemius and soleus is maximised in dorsiflexion. Traditional calculation of ankle joint moment based on a fixed centre
of rotation is acceptable only in exercises such as level walking with a limited range of motion about the neutral position. The
present model with a moving centre is particularly advised in exercises which take the joint nearer to the extremes of sagittal
motion. © 2002 Elsevier Science B.V. All rights reserved.
Keywords: Ankle complex; Muscle force; Lever arm; Four-bar linkage; Centre of rotation
1. Introduction
The ankle –subtalar complex plays a fundamental
role in the human locomotor system, being involved in
virtually every activity. This unit provides the rocker of
the shank with respect to the foot during the different
phases of the walking cycle [1]: (1) from the terminal
part of the swing phase, through the heel contact, until
the foot is flat on the ground, it controls the lowering of
the foot to the floor; (2) during the period in which the
foot remains flat on the ground and the shank advances, it controls the continued forward movement of
the body; (3) during the push-off phase, it allows the
generation of power for progression of the limb. As in
all other human joints, motion is guided by the osteoarticular and ligamentous structures and induced by the
forces and moments of the extrinsic muscles. Muscles
act by applying force through the muscle tendons with
instantaneous lever arms relative to the joint centre, the
tendons wrapping around bones and deflecting under
retinacula when necessary. The lever arm lengths are
* Corresponding author. Tel.: + 39-051-636-6522; fax: + 39-051636-6561.
E-mail address: [email protected] (A. Leardini).
measures of the ability of muscles to produce joint
torque in order to generate rotation and/or to resist
external forces. Any injury, lesion or neuromuscular
disorder of this complex system affects this interaction
between muscles, bones and ligaments and causes
degradation, instability or disability of locomotion.
To enhance understanding of disorders and of relevant conservative and surgical treatments, a better
knowledge of the physiological mechanics of the ankle
complex still remains a crucial issue. All the previous
models [2–6] assumed the ankle and subtalar joint to
be simple hinge and fully congruent joints, although
most of the recent experimental studies [7–13] strongly
suggest that this assumption should be questioned.
There is need for a mathematical model which can
describe the changing orientations of the muscle tendon
lines of action and positions of the instantaneous centre
of rotation. Particularly for the definition of the lever
arms of the main flexor and extensor muscles, there are
no previous modelling studies aimed at analysing the
mechanical effects of the retinacula constraining the
tendons at the ankle joint.
Lever arm lengths have been estimated in experimental studies in vivo and in vitro by indirect measurements. The ‘tendon excursion’ method has been largely
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A. Leardini, J.J. O’Connor / Gait and Posture 15 (2002) 220–229
used in cadaver studies [14– 17]. The instantaneous
length of the muscle lever arms is calculated as the ratio
between the instantaneous values of the tendon excursion and the joint rotation, the former being the distance moved by the tendon with respect to the
underlying bone during both concentric and eccentric
muscle contraction. This method of measurement accounts for the combined effect of (i) changing positions
of the rotation axis; (ii) the spanning of the tendon over
several joints; and (iii) action of retinacula, sheaths, or
bony structures influencing muscle and tendon course.
It has therefore been suggested that this length should
be called the ‘effective moment (lever) arm’ [15]. The
main concerns with this technique are the critical measurements of both tendon excursion and joint rotation,
and the underlying assumption that the line of action of
the muscle force and the axis of rotation are always
perpendicular.
Measurements of lever arm lengths for the Achilles
tendon and tibialis anterior were also performed in vivo
[18 – 21] on a series of sagittal plane magnetic resonance
images (MRI). Rugg et al. [18] found that moment arm
lengths decrease by approximately 20% for the Achilles
tendon and increase by approximately 30% for the
tibialis anterior when the ankle is moved from maximal
plantarflexion to maximal dorsiflexion. A 3.1 and 2.5%
average increase of the moment arm was obtained for
the two muscle groups, respectively when a fixed centre
was used. It was claimed that the averaged moment
arm lengths for these tendons were relatively unaffected
by the use of a moving centre as opposed to a fixed.
Maganaris et al. [19– 21] measured the changes in the
tibialis anterior and Achilles tendon moment arm
lengths with increasing muscle force. Sagittal-plane
MRI pictures were taken at rest and at maximum
voluntary contraction in several different joint positions. The authors concluded that a substantial increase
in the moment arms occurs because of the deformation
of the tissues. They did not discuss the patterns of
increasing of the tibialis anterior and Achilles tendon
moment arms with respectively increasing dorsi- and
plantarflexion of the ankle, nor the mechanical effects
of the retinacular constraints.
In understanding the mechanics of a human joint,
mobility studies can first elucidate the relationship of
kinematics to the geometry of the passive structures, i.e.
articular surfaces and ligaments. A previous extensive
investigation [22–25] has shown, by virtue of a computer-based model of ankle mobility in the sagittal
plane, that the articular surfaces and the ligaments
prescribe a unique envelope for the instantaneous positions of the axis of rotation [25]. The final goal of this
investigation was to analyse the changing directions of
the lines of action of contact, ligament and muscle
forces as the necessary preliminary information for the
estimation of the magnitude of the corresponding inter-
221
nal forces during activity. The aim of the present study
was to show these changing directions throughout the
flexion range and to analyse the corresponding effects
on muscle lever arms and forces during muscle
strengthening and gait exercises.
2. Materials and methods
2.1. The four-bar linkage (4BL) model of the ankle
joint complex
Experiments carried out by several authors on belowknee amputated specimens [22,24] have been later elucidated by a geometrical computer-based model [25]. In
summary, during passive flexion, the motion of the
ankle complex (calcaneus with respect to tibia/fibula
segment) occurs mostly at the ankle (tibiotalar) level.
This single degree-of-freedom (DOF) motion is guided
by the isometric rotation of the most anterior fibres of
the calcaneofibular (CaFi) and the tibiocalcaneal (TiCa)
ligaments [22]. The instantaneous centre of rotation
(IC), the point at which the two ligament fibres cross,
moves anteriorly and proximally during dorsiflexion.
The articular contact moves from the posterior part of
the tibial mortise in maximal plantarflexion to the
anterior part in maximal dorsiflexion. Rolling as well as
sliding occurs at the ankle joint during flexion: the talus
rolls forward while sliding backwards on the tibial
mortise during dorsiflexion, and vice versa during
plantarflexion.
The CaFi and TiCa ligaments are therefore essential
to control normal kinematics. The orientation of these
ligaments, the position of the instantaneous centre
about which dorsi/plantarflexion occurs and the position of the contact point all change relative to both
bony segments during flexion according to the ligament
linkage. Because of the changing positions of the IC
and of the points of tendon attachment and wrapping,
muscle lever arm lengths must also change with flexion.
2.2. The muscle action model
The sagittal model of ankle mobility was developed
to include the course of the main extensor and flexor
muscle groups and the extensor retinacular bands. Fig.
1 provides a schematic geometrical representation of
this model.
The geometry of this model in the neutral position
was defined by sets of anatomical parameters. The
course of the three extensor retinaculum bands and the
central areas of origin and insertion of the soleus and
tibialis anterior muscles were detected in three belowknee amputated specimens, using a stereophotogrammetric system and an anatomical landmark calibration
procedure [24]. Three-dimensional (3-D) positions of
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Fig. 1. The extended muscle action 4BL model in neutral position.
The courses (dashed) and lines of action (single-arrow segments) of
the muscles, the frictionless pulleys (grey circles), the extensor retinaculum bands (solid segments from the pulleys to the attachment areas)
are depicted. The band not included in the mechanical effect of the
extensor retinaculum is in dotted line. Muscle lever arm lengths
(double-arrow segments) are the orthogonal distances between muscle
force lines of action and instantaneous centre of rotation IC (empty
circle, at the cross point of the model ligaments CaFi and TiCa).
these landmarks in the neutral position were projected
in the sagittal plane. With respect to the gastrocnemius,
the two-dimensional (2-D) insertion point was taken at
the Achilles tendon attachment on the upper calcaneus,
whereas the origin was assumed as it was taken at knee
full extension in a geometrical sagittal model of this
joint [26,27]. This knee model showed how small is the
effect of knee flexion on the orientation of the gastrocnemius muscle in a tibial reference frame.
The muscle– tendon units (dashed lines) were modelled as single lines connecting relevant origins and
insertions approximated by single points. Each of the
main three extensor retinaculum bands was modelled as
a frictionless pulley (grey circles), around which the
tibialis anterior tendon wraps. The superior and inferior
lower bands were rigidly attached to the tibia/fibula
and calcaneus segments, respectively. In nature, the
inferior upper band of the extensor retinaculum runs
from the medial aspect of the distal tibia to the lateral
aspect of the calcaneus. In the present model, the
relevant pulley was taken to be rigidly attached to the
tibia/fibula segment, and the mechanical action of the
part of the band joining the pulley and the calcaneus
(dotted line) was ignored. The movement of the inferior
lower pulley was calculated from 4BL kinematics.
In such a multiple attachment/wrapping mechanism,
when the tibia/fibula is taken as the fixed body and the
talus/calcaneus as the moving body, the effective line of
action of a pulling force joins the relevant extreme
connection points. The line of action of the corresponding muscle force which spans the ankle joint therefore
coincides with the tendon course between the upper and
lower inferior retinaculum (single-arrow segment on the
tendon course). The lever arm length of the tibialis
anterior muscle (double-arrow segment) was therefore
calculated as the orthogonal instantaneous distance between this muscle force line of action and the location
of the IC at any joint position. Lever arm lengths
(double-arrow segments) of the gastrocnemius and
soleus muscles were simply taken as the instantaneous
distances between relevant muscle lines of action (single-arrow segments) and the location of the IC.
A diagrammatic sketch from the computer animation
based on this model is given in Fig. 2. The changing
positions of the muscles, retinaculum bands, ligaments,
and relevant centre of rotation and common normal in
the sagittal plane are depicted at three characteristic
joint positions.
Because both the lines of action of the muscle tendons and the centre of rotation change position during
flexion as guided by the ligament linkage, the lever arm
lengths of the three muscles change accordingly. The
lever arms of the tibialis anterior and gastrocnemius are
depicted in the figure. Because the centre of rotation
Fig. 2. A diagrammatic sketch of the mechanism of muscle leverage at the ankle complex in the sagittal plane as predicted by the muscle action
development of the 4BL model. Lines of action (single-arrow segments) of the three muscles and lever arm lengths (double-arrow segment) of the
tibialis anterior and gastrocnemius are shown at 20° plantarflexion (a), at neutral (b), and at 10° dorsiflexion (c) positions. Positions of the
instantaneous centre (IC, empty circle) and of the contact point identified by the common normal (CN, dash – dotted line) are also reported.
A. Leardini, J.J. O’Connor / Gait and Posture 15 (2002) 220–229
223
Fig. 3. Calculated lever arm lengths (solid lines) of the main flexor and extensor muscles against ankle dorsi/plantarflexion angle. Corresponding
patterns for gastrocnemius (dash –dotted), soleus (dotted) and tibialis anterior (dashed) muscles calculated for a fixed-hinge ankle joint are
superimposed.
moves forwards and upwards during dorsiflexion, the
lever arm length of tibialis anterior becomes shorter as
the ankle moves from maximal plantar- to maximal
dorsiflexion, and vice versa for the two flexor muscles.
For comparison, lever arm lengths were also calculated
as though the ankle joint were a hinge with axis of
rotation fixed at the location assumed in the neutral
position.
2.3. The muscle strengthening exercises
To study the effect of the combined mechanical
action of the muscle and of the external forces to be
resisted during activity, muscle strengthening exercises
were first simulated using the computer model
configured with the same set of geometrical parameters.
The tibia was thought to be held oblique at 45° from
the horizontal facing up and down, respectively for the
strengthening of the extensor and flexor muscles. A unit
load was thought to be hung to the posterior aspect of
the calcaneus and the anterior aspect of the talus,
respectively. A 23° plantar /25° dorsiflexion range of
motion was considered. Directions of the lines of action
of the muscle forces and of the external load throughout the flexion range were calculated from the model.
2.4. The gait analysis test
A gait analysis test was carried out on an adult male
volunteer (582 N weight, 1.74 m height), with leg
dimensions roughly similar to that of one of the specimens analysed. Trajectories of anatomical landmarks
on the shank and on several foot segments necessary to
define the parameters of the model were tracked by a
stereophotogrammetric system (ElitePlus, BTS, Milan)
and a standard protocol [28,29]. Flexion angle and
moment at the ankle were also calculated accordingly
[29]. Ground reaction force (GRF) was measured by
means of a force platform (Kistler, Switzerland). Electromyography (Telemg, BTS, Milan) was also used to
assess muscle activation (on–off) timing, according to
an established algorithm [30]. Kinematic, electromyographic and force data were collected simultaneously.
Sagittal projection of the instantaneous position of
the GRF vector was expressed in the same tibial reference frame as the sagittal mechanical model and considered to be the external load resisted by the relevant
muscle force. Patterns of change in IC position and
muscle line of action orientation for every joint flexion
angle were calculated. Muscle action was assumed to
balance the joint moment induced by the GRF, therefore flexors act when GRF passes anterior to IC,
extensors when GRF is posterior to IC.
3. Results
3.1. Prediction of muscle le6er arm lengths
Fig. 3 shows the calculated lever arm lengths of the
three muscles (solid lines) plotted against corresponding
flexion angle of the ankle. A 23° plantar /25° dorsiflexion range of motion was imposed according to the
relevant motion performed by one of the specimens
analysed. With the model configured as in Fig. 1, both
the anterior and proximal displacement of the IC was
approximately 7 mm.
The figure shows that the lever arm of tibialis anterior increases from approximately 3.1 cm in dorsiflexion
to 3.8 in plantarflexion, an increase of 23%. This trend
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A. Leardini, J.J. O’Connor / Gait and Posture 15 (2002) 220–229
was exactly opposite to that obtained from the fixedaxis model which predicted a decrease from a maximum
of 3.8 to a minimum of 3.4 cm, 11%. The gastrocnemius has a maximum lever arm length of 5.4 cm at 7°
dorsiflexion, with a total lever arm length excursion of
1.0 cm. The corresponding figures for the fixed-axis
model are 5.3 cm at 0° flexion and 0.6 cm excursion,
resulting in a 40% decrease of lever arm length excursion. The soleus has a maximum lever arm length of 5.3
cm at 0° flexion, with a total lever arm length excursion
of 0.7 cm. The corresponding figures for the fixed-axis
model are 5.3 at 7° plantarflexion, 0.9 cm excursion.
Between 910° flexion from the neutral position, all
these differences are small.
The calculation of lever arm lengths with sets of
geometrical parameters taken from other specimens
produced similar results.
3.2. Forces at the ankle in muscle strengthening
exercises
Fig. 4 shows the lever arm length and the corre-
sponding muscle force of the tibialis anterior muscle for
the simulation of the strengthening exercise.
Fig. 4a shows that, in the present model of moving
IC, the lever arm of the tibialis anterior is larger in
maximal plantarflexion than in maximal dorsiflexion,
and that the contrary occurs for the external load.
These two contributions combine in Fig. 4b. It shows
that in maximal dorsiflexion the value of the muscle
force necessary to resist an external load with a moving
centre joint is approximately 41% bigger than that
obtained with a fixed centre, but approximately 22%
less in maximal plantarflexion.
In maximal plantarflexion, the forces in the soleus
and gastrocnemius muscles necessary to resist independently the external load in a moving centre joint was
found, respectively to be approximately 20 and 21%
bigger than that obtained with a fixed centre, but 4 and
5% less in maximal dorsiflexion. The calculation of
lever arm lengths and simulation of the two strengthening exercises was repeated with sets of geometrical
parameters taken from two other specimens with similar results.
Fig. 4. Lever arm lengths of the tibialis anterior force and external load (a) and the magnitude of the tibialis anterior muscle force (b) necessary
to counterbalance a unit external load in muscle strengthening exercise. The patterns for the present moving IC are reported with solid lines.
Corresponding patterns for the tibialis anterior (dashed) and external load (dotted) for a fixed-hinge ankle joint are superimposed.
A. Leardini, J.J. O’Connor / Gait and Posture 15 (2002) 220–229
225
Fig. 5. Time history of muscle activation (grey bars) from EMG, of dorsi/plantarflexion angle (solid line, [°]) and of dorsi/plantarflexion moment
associated to the GRF (dashed line, [% body weight × height]) at the ankle from the gait analysis test.
3.3. Forces at the ankle during gait
In Fig. 5, muscle action timings of the gastrocnemius
and tibialis anterior muscles (grey bars) are superimposed on flexion angle and flexion moment of the ankle
joint over a gait cycle. Figs. 3 and 4 showed that the
mechanical advantage increases in some joint positions
and decreases in others. The joint positions in which
the two muscle groups fire are exactly those in which
they were predicted to be mechanically advantaged.
Fig. 6a and b show lever arm lengths and the necessary force magnitudes for both the moving (solid lines)
and fixed (dashed lines) centre joint model. A maximum
advantage of approximately 15 and 4% was found,
respectively, for the tibialis anterior and gastrocnemius
muscle forces, accounted for by the combined action of
a shorter lever arm for the GRF and a longer lever arm
for the muscle forces.
4. Discussion
4.1. Main results and rele6ance of the present work
In the present study, the lever arm length variations
for the gastrocnemius, soleus, and tibialis anterior muscles are predicted in a systematic way by an extension
of a previously validated 4BL model of ankle mobility
[24,25]. The previous study showed how a 4BL model
defines the changing geometry of the joint. It was
demonstrated that the ligament-based linkage guides
the axis of rotation to move anteriorly and proximally
during dorsiflexion. The present study now provides
directions of the muscle forces and the relevant lever
arm lengths, through the guided relative motion of the
two bones. It is demonstrated here that, because of the
changing positions of both muscle lines of action and
IC, the lever arm lengths change significantly with
flexion. Lever arms of the flexor muscles are maximised
in dorsiflexion, that of the extensor muscle are maximised in plantarflexion.
The path of IC displacement makes the action of the
tibialis anterior advantaged in plantarflexion, the action
of the gastrocnemius and soleus advantaged in dorsiflexion. The mechanical benefit is further improved by
considering that the action of the external forces to be
resisted during activities may be inversely advantaged.
The combination of the two factors makes the action of
the tibialis anterior even more advantageous in ankle
positions close to maximal plantarflexion and vice versa
for the two flexor muscles.
The ability of a muscle to produce joint rotation in
the presence of external forces depends on the relevant
lever arm length, defined as the perpendicular distance
from the axis of joint rotation to the line of action of
the muscle–tendon unit force. Knowledge of instantaneous muscle leverage is therefore fundamental in human movement analysis and joint modelling. No
previous mathematical models of the human ankle have
addressed the mechanical effect of the retinacula or
have presented the changing positions of the instantaneous centre of rotation, the directions of the lines of
action of the main flexor and extensor muscles and the
resulting lever arm lengths.
4.2. Limitations of the present model
The present model demonstrates the mechanical effects of the extensor retinaculum action in constraining
the tibialis anterior tendon. Several simplifications were
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still made here. The elastic stretching of the extensor
retinacula as the tibialis anterior contracts was not
considered. This effect would produce an increase of
the muscle lever arm [21]. The present model also does
not completely consider the mechanical action of the
inferior upper band of the extensor retinaculum. However, it is reasonably expected that this area of wrapping of the tibialis anterior tendon does not move with
respect to the tibia during ankle flexion, as it does the
inferior lower band, rigid with the foot.
Further limitations of the present study are associated to the 2-D nature of the model and to the assumed
rigidity of the joint structures. Particularly, this sagittal
model does not include the significant medio-lateral
component of the tibialis anterior tendon course and
the associated further effect in inverting/everting the
foot. Although foot and ankle have 3-D functions, the
activities analysed occur mainly in the sagittal plane,
and the present 2-D model has elucidated the key
mechanical features of the flexors and extensor muscles.
Extension of the model to 3-D is still under development and relevant results will enhance present knowledge. The effects of deformable articular surfaces and
extensible ligament fibres have been extensively investi-
gated for the knee joint [31,32]. It has been demonstrated that although tissue deformation is critical for
the calculation of ligament forces, it does not affect
much calculation of muscle forces.
In the present determination of the forces transmitted
by the structures at the ankle joint, a unique solution
with a single muscle action has been only considered
for any loading situation. Possible isometric simultaneous contractions of extensor and flexor muscles are in
addition to those required to balance the external loads,
and form systems of muscle, ligament and compressive
contact forces with no resultant. Arbitrary values of
these systems of forces can be added to the single
muscle solution to give further solutions valid over the
entire range of flexion. The present single muscle solutions therefore represent the minimal response of the
joint to external load, at least in terms of muscle and
contact force.
The predicted mechanical advantage during muscle
activity must be considered here as only associated to
the production of movement and as only accounted for
the displacement, though small, of IC. The present
considerations have no reference to any features associated to the production of force at the muscle–tendon
units, such as the muscle–tendon interaction, the
length–tensions and force–velocity relationships, and
to the muscle fibre recruitment. Whatever would be the
tension force developed at the muscle– tendon unit in a
gait cycle, the associated generated torque is maximised
by the pattern of motion of the centre of rotation.
There could be situations, for example, in which the
muscle force generated and the lever arm are not optimised at the same joint angle.
Finally, the present study is limited by the small
number of experiments. However, the few observations
on specimens confirmed a consistent pattern of anteroproximal translation of IC during dorsiflexion, the
main cause for muscle lever arm changes. The only one
volunteer analysed showed patterns of muscle activation and joint motion well within standard gait analysis
reports.
4.3. Choice of muscle fulcrum point
Fig. 6. Lever arm lengths (a) and magnitudes of the relevant forces
(b) necessary to counterbalance the external GRF, expressed for both
the 4BL (solid lines) and fixed centre of rotation (dashed lines) joint
models throughout the stance phase of a gait cycle.
Any system of forces, and in particular the system of
muscle, ligament and articular contact forces transmitted across a joint can always be reduced to a single
resultant force on any chosen line of action, plus a
couple. The magnitude of the resultant force is given by
the vector summation of the muscle, ligament and
articular contact forces and the moment of the couple is
given by the summation of the moments of the forces.
In moment equilibrium equations, any point (in 2-D)
or axis (in 3-D) can be taken as the fulcrum point about
which torque can be derived for every force acting on
the joint (muscular, external, ligament and contact). It
A. Leardini, J.J. O’Connor / Gait and Posture 15 (2002) 220–229
is recognised that the centre of rotation moves with
respect to both bones during rotation in several human
joints and it is therefore not the most simplest point for
the calculation of joint moments in in vivo movement
analysis tests. Although this centre is also not easily
determined, it is preferred to the even more difficult
evaluations of other characteristic points such as the
articular contact or the centre of curvature. Effective
moment arm for the muscle forces can be estimated as
the partial derivative of the tendon excursion with
respect to the joint rotation as in previous studies. The
external force, typically the ground reaction force, has
such a long moment arm that the corresponding moment is not too much affected by movement of the
centre of rotation. The ligament and contact forces are
ignored in moment equilibrium equations because they
pass through the centre of rotation in frictionless joints
[25,26]. A more accurate representation of the instantaneous positions of the centre of rotation is therefore
certainly fundamental in both routine human movement analysis and in modelling studies of joint mechanics. The present study is the first attempt to define
changing positions of the lines of action of the internal
forces at the ankle joint in a systematic way.
In a simplified configuration with only the external,
the muscular and the contact forces, this latter can be
easily determined from the force equilibrium. The magnitude of this force was found to be very similar to the
sum of the magnitudes of the external and muscular
forces because their orientations are almost parallel
throughout the flexion range, and therefore not reported. The present anatomically based geometrical
model is also able to provide the changing directions of
the forces transmitted by muscles, ligaments and articular surfaces. These are fundamental data for further
mechanical analyses aimed at calculating the magnitude
of these forces.
4.4. Comparison with pre6ious works
The present predictions of lever arm length changes
are in general agreement with the results reported
by Spoor et al. [15]. The present results are in disagreement with the other previous studies based on MRI
measurements, where the position of the instantaneous
centre of rotation obtained by Reuleaux’s graphical
method [18–21] is liable to significant errors [33]. Using
this latter technique, the patterns of lever arm changes
reported for both a fixed and a moving centre
of rotation were found to be very similar. These studies
in fact failed to show a systematic path of motion for
the centre of rotation. These patterns of lever arm
changes are instead similar to those reported here when
simulating a fixed centre of rotation (dashed lines in
Fig. 3), demonstrating the general weakness of this
technique.
227
4.5. Mechanics of the ankle in the gait cycle
Though estimated in the sagittal plane only, the
prediction of muscle action during gait was found to be
attractive. When looking at the mechanics of the gait
cycle, the GRF is the external load to be balanced. IC
moves in the same direction of the displacement of
GRF vector during the stance phase of walking, causing a generally smaller lever arm and a corresponding
smaller generated torque to be resisted than what it
would be for a fixed centre of rotation. These changes
affect disadvantageously GRF action during the walking cycle. The firing muscles generally lie opposite to
the GRF and therefore are inversely advantaged by the
movement of IC. Action of the flexor muscles is maximised in ankle dorsiflexion, action of the extensor in
plantarflexion, exactly those joint positions in which
the two muscle groups fire in the gait cycle. The mechanical advantage of the gastrocnemius (Fig. 6b) is a
little smaller than corresponding tibialis anterior advantage because of the much larger lever arm of the
GRF at push-off phase (Fig. 6a). A slightly smaller
muscle force is therefore necessary to resist GRF and
produce joint movement, and a smaller contact force is
also expected. This advantage can elucidate the role of
the moving centre in reducing muscle forces and therefore joint loading during level walking. Although further gait analysis tests would be necessary to confirm
these observations and in vitro tests would better support experimentally these results, the present observation of a mechanical, though small, advantage at the
ankle joint in the gait cycle supports the assumptions
underlying the formulation of the model.
4.6. Final recommendations
Comparison of muscle force estimation in strengthening (Fig. 4) and gait (Fig. 6) exercises points out a
difference in the mechanical advantage of the muscle
action associated to the moving centre of rotation. In
the stance phase of level walking, the advantage may
be marginal being limited to a maximum of 4% for the
gastrocnemius, although 15% for the tibialis anterior.
In the muscle strengthening exercise, the tibialis anterior can reach an advantage of 22%. Fig. 3 reveals that
this difference is accounted for the much larger range
of motion imposed in muscle strengthening (23° plantar/25° dorsiflexion) than that observed in the gait cycle
(10° plantar/10° dorsiflexion).
In the clinical context, it is here advised that careful
restoration of the original geometry of the ligamentous
and muscle–tendon apparatus would allow not only a
more physiologic pattern of joint kinematics, but also a
more physiologic pattern of joint loading. The present
model would advance the estimation of the resulting
external moment and power at the ankle joint in move-
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ment analysis and the estimation of the internal forces
at the single anatomical structures. A more accurate
estimation of the effects of muscle– tendon unit lengthening can certainly be obtained by the present geometrical model for a better assessment of the relevant
surgical interventions.
5. Conclusions
This study has shown how a previous model of ankle
mobility enhanced with a model of the mechanical
effect of the extensor retinacula is effective in a systematic prediction of flexor and extensor lever arm lengths
at the human ankle joint. Displacement of the centre of
rotation significantly affects estimation of the muscle
lever arm lengths. Traditional calculation of ankle joint
moment based on a fixed centre of rotation model are
acceptable only in exercises with a limited range of
motion near the neutral position (gait). A fixed centre
of rotation model for the ankle should not be used in
the analysis of human exercises which take the joint
nearer extremes of motion in the sagittal plane (muscle
strengthening but also stair climbing/descending,
raising from a chair, deep squat, etc.). In these latter,
the present 4BL mechanical model is particularly advised, pending development of a 3-D anatomical model
of the joint system.
Acknowledgements
This work was supported by the Italian Ministry of
Health Care and The Arthritis Research Campaign of
Great Britain.
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